| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for positive correlation |
| Difficulty | Standard +0.3 This is a straightforward application of Spearman's rank correlation coefficient with clear data, requiring ranking, calculating the coefficient using the standard formula, and performing a one-tailed hypothesis test at 1% significance. While it involves multiple steps, each is routine for S3 students with no conceptual challenges or novel insights required—slightly easier than average due to its mechanical nature. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank |
| Position | A | B | \(C\) | D | \(E\) | \(F\) | G |
| Distance from inner bank \(b \mathrm {~cm}\) | 100 | 200 | 300 | 400 | 500 | 600 | 700 |
| Depth \(s \mathrm {~cm}\) | 60 | 75 | 85 | 76 | 110 | 120 | 104 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rank depths against distances | M1 | Attempt to rank the depths against distances |
| Calculate \(d\) for their ranks | M1 | Attempting \(d\) for their ranks; must use ranks |
| \(\sum d^2 = 8\) | M1A1 | Attempting \(\sum d^2\); sum of 8 (or 104 for reverse ranking) |
| \(r_s = 1 - \frac{6\times8}{7\times48}\) | M1 | Use of correct formula with their \(\sum d^2\) |
| \(= \frac{6}{7} = 0.857142...\) | A1 | awrt \((\pm)0.857\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0: \rho=0,\; H_1: \rho>0\) | B1 | Both hypotheses in terms of \(\rho\); \(H_1\) must be one-tail and compatible with ranking |
| Critical value at 1% level is \(0.8929\) | B1 | cv of \(0.8929\) (accept \(\pm\)) |
| \(r_s < 0.8929\) so not significant evidence to reject \(H_0\) | M1 | Correct statement relating their \(r_s\) with cv; cv must satisfy \( |
| The researcher's claim is not correct (at 1% level) / insufficient evidence for researcher's claim / insufficient evidence that water gets deeper further from inner bank | A1ft | Correct contextualised comment; must mention "researcher" and "claim" or "distance (from bank)" and "depth (of water)"; use of "association" is A0 |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rank depths against distances | M1 | Attempt to rank the depths against distances |
| Calculate $d$ for their ranks | M1 | Attempting $d$ for their ranks; must use ranks |
| $\sum d^2 = 8$ | M1A1 | Attempting $\sum d^2$; sum of 8 (or 104 for reverse ranking) |
| $r_s = 1 - \frac{6\times8}{7\times48}$ | M1 | Use of correct formula with their $\sum d^2$ |
| $= \frac{6}{7} = 0.857142...$ | A1 | awrt $(\pm)0.857$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \rho=0,\; H_1: \rho>0$ | B1 | Both hypotheses in terms of $\rho$; $H_1$ must be one-tail and compatible with ranking |
| Critical value at 1% level is $0.8929$ | B1 | cv of $0.8929$ (accept $\pm$) |
| $r_s < 0.8929$ so not significant evidence to reject $H_0$ | M1 | Correct statement relating their $r_s$ with cv; cv must satisfy $|\text{cv}|<1$ |
| The researcher's claim is not correct (at 1% level) / insufficient evidence for researcher's claim / insufficient evidence that water gets deeper further from inner bank | A1ft | Correct contextualised comment; must mention "researcher" and "claim" or "distance (from bank)" and "depth (of water)"; use of "association" is A0 |
---\begin{enumerate}
\item A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at seven positions. The results are shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Position & A & B & $C$ & D & $E$ & $F$ & G \\
\hline
Distance from inner bank $b \mathrm {~cm}$ & 100 & 200 & 300 & 400 & 500 & 600 & 700 \\
\hline
Depth $s \mathrm {~cm}$ & 60 & 75 & 85 & 76 & 110 & 120 & 104 \\
\hline
\end{tabular}
\end{center}
(a) Calculate Spearman's rank correlation coefficient between $b$ and $s$.\\
(b) Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a $1 \%$ level of significance.\\
\hfill \mbox{\textit{Edexcel S3 Q4 [10]}}